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Math Help - fermat...or not fermat?

  1. #1
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    fermat...or not fermat?

    proove that if a prime number is represented by this phormula p=(2^n)+1 with n>0 then n is a power of 2... it's all for you...
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  2. #2
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    Quote Originally Posted by Aglaia View Post
    proove that if a prime number is represented by this phormula p=(2^n)+1 with n>0 then n is a power of 2... it's all for you...
    The important fact is that, we can factor:
    x^n+y^n where n is odd.

    Now, assume,
    2^n+1 is prime where n is divisible by odd.
    Then,
    2^{(2k+1)j}+1
    Thus,
    (2^j)^{2k+1}+1
    Can be factored as,
    (2^j+1)(2^{2kj}-2^{(2k-1)j}+2^{(2k-2)j}-...+2^{2j}-2^j+1)
    It has a proper nontrivial factorization, thus it cannot be prime.
    Thus, n cannot be divisble by odd number that is, 2^m
    Thus,
    2^{2^m}+1
    Which were studied by Fermat (my favorite mathemation).
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